Multiple choice

Solve the following quadratic equation using quadratic formula: $a^2x^2+(a^2-b^2)x-b^2=0$

  1. $\left \{ -1, \dfrac{b^2}{a^2} \right \}$
  2. $\left \{ 1, \dfrac{b^2}{a^2} \right \}$
  3. $\left \{ -1, \dfrac{-b^2}{a^2} \right \}$

  4. None of these
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A Correct answer
Explanation

The equation a^2x^2 + (a^2 - b^2)x - b^2 = 0 can be factored as (a^2x - b^2)(x + 1) = 0. The roots are x = -1 and x = b^2/a^2.

AI explanation

For the equation a^2x^2 + (a^2 - b^2)x - b^2 = 0, we factor the quadratic expression by splitting the middle term. We rewrite the middle term to get a^2x^2 + a^2x - b^2x - b^2 = 0. Factoring by grouping gives a^2x(x + 1) - b^2(x + 1) = 0, which results in (a^2x - b^2)(x + 1) = 0. Setting the factors to zero yields the roots x = -1 and x = b^2/a^2.

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